The majority of people move to Japan to either teach English, or be employed in a role that requires dealing with people who speak English, or a language other than Japanese.
From Wikipedia:
"In 2010, the number of foreigners in Japan was 2,134,151. There was 230,552 Brazilians who are mostly of Japanese descent, 687,156 Chinese and 565,989 Koreans. Koreans, Chinese and Brazilians account for about 69,5% of foreign residents in Japan."
So, you're wrong right off the bat. No surprise considering your profile shows you have never been to Japan.
However, does the Japanese government, or Japanese companies ever sponsor skilled workers from overseas to work in Japan in a normal job that is not targetted specifically at foreigners?
For example, if a registered Paramedic with a degree in Paramedic Science from the UK wanted to work in Japan, would they be able to find a hospital or ambulance provider willing to sponsor them, and if so, would the immigration authorities be likely to approve a visa?
For jobs that require licenses (such as the above) the license needs to come from Japan, not abroad. Same holds for doctors and lawyers.
Skilled laborers do exist in Japan. That's why there is a skilled labor work visa. However, Japan is notorious for importing such foreign work on a short-term basis almost akin to slave labor and in illegal fashion.
Japanese Trainee Program Accused of Exploiting Migrant Workers - NYTimes.com
Recently, it opened its door to nurses from SE Asia, but that has pretty much flopped because they never provided any language training and then complained about the poor language skills, and almost none could pass the nursing exam.
Japan Keeps a High Wall for Foreign Labor (Published 2011)
Coffeesan said:
I've heard Japan really wants immigrants at the moment
Wants? No, not outwardly. Needs? Yes, definitely, and for the reasons you cited.
Infection said:
Well, speaking Japanese to a near native level is an obvious requirement,
Look again. JLPT2 (or its current equivalent) is often a minimum acceptable level, far from "near native level".
but I get the impression that for many jobs in Japan, foreign qualifcations are often not considered as equivalents to the Japanese qualifications and that in many cases foreigners would have to retrain from scratch?
Why should a doctor or lawyer have the same credentials in a foreign land? Do you even have the slightest clue, for example, how a Japanese lawyer or doctor gets their training? (Psst, it's not like in the UK.)